ࡱ> y bjbj {{10 0 4 a\a\a\ސ$EQa\YLa\a\a\M;U___a\pސ_a\ސ__ lRj+]v*ʐk0^6TRRxa\a\_a\a\a\a\a\_a\a\a\a\a\a\a\a\a\a\a\a\a\a\a\a\0 P: ALGEBRA I FORMULAS AND FACTS FOR EOC Commutative Property:a + b = b + aAssociative Property:(a + b) + c = a + (b + c)Order of Operations:PEMDASPercent of Change: EMBED Equation.3  or  EMBED Equation.3 Distributive Property:Distribute (multiply) a term to each individual term inside the parentheses. Be careful of negative and positive values. -3y (5 + x - 2y) = (-3y)*5 + (-3y)*x + (-3y)*(-2y) = -15y-3yx + 36y2Combining Like Terms:To combine like terms, the variables and exponents must be same. 3x2 + 4x 6 + 6x + 14 5x2 = (3 - 5)x2 + (4 + 6)x +( 6 + 14) = -2x2 + 10x + 8Solving equations:CALCULATOR: Left Side = Y1. Right Side = Y2. [GRAPH]. Find Intersection: [2nd], [Trace], [5:Intersect], [ENTER], [ENTER], [ENTER] By HAND: Use SADMEP cancel operationsIndependent Variable:x-valuesDependent Variable: y-valuesDomain:All x-values used an equation, function, or graph (left and right)Range:All y-values used an equation, function, or graph (up and down)Greatest Common Factor (GCF)Largest Integer into all numbers and smallest exponent of variable Exp: 10x2yz4 and 15x5y3; GCF = 5x2yLeast Common Multiple (LCM)Smallest integer multiply to equal, largest exponent each variable Exp: 10x2yz4 and 15x5y3; LCM = 30x5y3z4Slope: between (x1, y1) and (x2, y2) EMBED Equation.3 or  EMBED Equation.3 Parallel Lines:Same slopePerpendicular Lines:Opposite (Negative) reciprocal slope.Slope Intercept form:y = mx + b m = slope and b = y-interceptPoint-Slope Form:y y1 = m (x x1) point: (x1, y1) and m = slopeStandard Form:Ax + By = C GCF of A, B, and C = 1 NO FRACTIONS A is positiveHorizontal Line:Equation: y = #; Zero (No) SlopeVertical Line:Equation: x = #; Undefined SlopeDirect Variation:y = kx ; y varies directly with x; multiply from X to Y or vice versaMidpoint Formula: between (x1, y1) and (x2, y2) EMBED Equation.3 Matrices: rows go across, columns go down.CALCULATOR: Create - [2nd], [Matrix], EDIT, Select a Matrix to use ([A], [B], etc), Input Rows and Columns, Input Elements Use Operations [2nd], [Matrix], NAMES, Select a Matrix ([A], [B],)Solving a system of equations: GRAPHING: Solve for slope intercept form Plug into y = and find the intersection of the lines MATRIX: Write both equations in STANDARD FORM of lines  EMBED Equation.3  MATRIX =  EMBED Equation.3  Plug coefficients and constants matrix and perform RREF operation. If bottom row is 0 1 #, the last column is your answer. If bottom row is 0 0 1, there is no solution. If bottom row is 0 0 0, there are infinitely many solutions.Graph Inequalities:Step #1: Solve for slope intercept form. If you multiply or divide by a negative, then flip direction of inequality. Step #2: Solid Line when e" or d" AND Dotted Line when > or < Step #3: Shade Up (right) when e" or > AND Shade Down (left) when d" or < Distance Formula: between (x1, y1) and (x2, y2)DISTANCE = EMBED Equation.3 ; Draw a right triangleQuadratic Formula: ax2 + bx + c = 0  EMBED Equation.3 Pythagorean Theorem:a2 + b2 = c2Exponential Functions:y = abx b = base or pattern of multiplication, a = initial valueExponential Growth: (Increases/ Appreciates)y = a(1 + r)x a = initial value, r = rate of percent increase (4.5%; r = 0.045)Exponential Decay: (Decrease/ Depreciates)y = a(1 - r)x a = initial value, r = rate of percent decrease (5.7% r = 0.057) Common Shapes Area and/or Perimeter Formulas Perimeter of a Figure: add up all sidesArea of a Circle: A = (r2 Circumference of Circle: C = 2(r SHAPE \* MERGEFORMAT Area of a Rectangle: A = l*w Perimeter of a rectangle: P = 2l + 2w SHAPE \* MERGEFORMAT Area of Triangle: A = bh Perimeter of a triangle: P = s1 + s2 + s3  SHAPE \* MERGEFORMAT Area of a Square: A = s2 Perimeter of a square: P = 4s  SHAPE \* MERGEFORMAT  Area of a Trapezoid: A = (b1 + b2)h SHAPE \* MERGEFORMAT  Volume of a Cylinder: V = (r2hCALCULATOR COMMANDS Graphing: [Y=] enter in the equation, [ZOOM], [6: ZStandard] If the graph is not shown, then the change the window by: Make sure xmin < xmax and ymin < ymax [WINDOW] and adjust YMAX (see farther up) XMIN (see more left) XMAX (see more right) YMIN (see farther down)  To find the MAXIMUM VALUE: [2nd], [TRACE], [4] (maximum) Left Bound: move the cursor to the left of the maximum (top of hill) ENTER Right Bound: move the cursor to the right of the maximum (top of hill) ENTER Guess: move the cursor to the maximum (top of the hill) ENTER  To find the MINIMUM VALUE: [2nd], [TRACE], [3] (minimum) Left Bound: move the cursor to the left of the minimum (bottom of valley) ENTER Right Bound: move the cursor to the right of the minimum (bottom of valley) ENTER Guess: move the cursor to the minimum (bottom of valley)ENTER  To find the ROOTS/ ZEROS/ X-INTERCEPTS: [Y =] make Y1 = Equation and Y2 = 0 [GRAPH], [2nd], [TRACE], [5] (intersect) Move cursor to intersection [ENTER], [ENTER], [ENTER] To find X when Y = #: [Y =] make Y1 = Equation and y2 = #, [GRAPH], [2nd], [TRACE], [5] (intersect) Move cursor to intersection ENTER, ENTER, ENTER Make sure that your window shows the intersection To find Y when X = #. Option #1: [2nd], [TRACE], [1] (value), X= #, [ENTER] Option #2: [2nd], [WINDOW] let TblStart = #, [2nd], [Graph] To find INITIAL VALUE or Y-INTERCEPT, look when x = 0. LINES OF BEST FIT and LINEAR REGRESSION and PREDICTION EQUATIONS INPUTTING DATA: STAT -> EDIT L1 X VALUES, L2 - Y VALUES Make sure that rows represent points/ ordered pairs MAKE A SCATTER PLOT: 2ND, STAT PLOT [Y=], ENTER TURN STAT PLOT ON (ENTER) TYPE: HIGHLIGHT FIRST GRAPH X-LIST: L1 Y-LIST: L2 SEE THE ENTIRE SCATTER PLOT: WINDOW XMIN: # < smallest number in L1 XMAX: # > biggest number in L1 YMIN: # < smallest number in L2 YMAX: # > biggest number in L2 FINDING THE LINE OF FIT: [STAT] -> Scroll to CALC 4 [LINREG(ax + b)] a = slope (m) and b = y-intercept VARS, Y-VARS, FUNCTION, Y1, [ENTER] GO TO [Y =]; You can now see the equation of the line GRAPH; You can see the stat plot and the line together Now have an equation to help predict values by Searching the TABLE: 2ND, WINDOW, TblStart = value you want 2ND, GRAPH [TABLE] Finding a specific VALUE: 2ND TRACE [CALC] 1 - ENTER TOUGH VERBAL TRANSLATIONS 1) 4 less a number: n 46) 5 more a number: 5 + n 2) 4 less than a number: 4 n7) 5 is more than a number: 5 > n3) 4 is less than a number: 4 < n8) 5 more than a number: n + 54) a number is at least 10: n e"109) a number is at most 10: n d" 105) 4 less of the difference of a number and 5: 4  (n  5)10) 2 more than the sum of a number and 12: (n + 12) + 2 Chapter 9 Factoring Review Section 9.1: Is a number prime or composite? Find the Prime Factorization: All Prime numbers that multiply together to equal a number. Draw your tree. Example: 180 = 2*2*3*3*5Find the Greatest Common Factor (GCF) between numbers: pair up all common prime factors of the numbers and multiply together Example: GCF = 2*3 or 6 72 = 2 * 2 *2 * 3* 3 36 = 2 * 2 * 3* 3 42 = 2 * 3 * 7 Find the Greatest Common Factor (GCF) between monomials: find the GCF of the coefficients and then for any common variable pick the smallest exponent for each variable Example: GCF = 2*x2*y3 = 2x2y3 6x2y6 = 2*3*x2*y6 32x3y4 = 2*2*2*2*x3*y4 10x5y3 = 2*5*x5*y3 FACTORING TECHNIQUEEXAMPLES2 or more termsGREATEST COMMON FACTOR (GCF): (1) Find the GCF of all terms of the polynomial, (2) Divide the polynomial by the GCF, and (3) Write the factors as the GCF times polynomial from step 2. 3x3 + 6x2 + 15x (1) GCF = 3x (2) (3x3 + 6x2 + 15x)/(3x) = x2 + 2x + 5 (3) = 3x(x2 + 2x + 5)2 termsSpecial Case: DIFFERENCE OF SQUARES: a2 b2 = (a + b) (a b)4x2 25 (2x)2 (5)2 (2x 5) (2x + 5)3 termsax2 + bx + c Step #1:Greatest Common Factor of a, b, and c Step #2: Factor-Sum Tree to find a pair M and N Multiply to equal = product of a and c Add to equal = b Step #3: Split the middle term, bx = Mx + Nx Step #4: Factor by Grouping (See below) x2 + 11x + 24 3*8 = 24 and 3 + 8 = 11 x2 + 3x + 8x + 24 = (x + 3)(x + 8)6x2 x 2 3*-4 = -12 and 3 + - 4 = -1 = 6x2 + 3x 4x 2 = 3x(2x + 1) 2(2x + 1) = (3x 2) (2x + 1)4 termsFactor by Grouping: Factor the 1st 2 terms and the last 2 terms separately by their respective GCFs. ax + bx + ay + by 3x2 6x + 5x 10 = (3x2 6x) + (5x 10) GCF = 3x GCF = 5 = 3x(x 2) + 5(x 2) = (3x + 5) (x 2) Hint: If x (a - b) + y(-a + b), then make a subtraction for y x (a - b) y(a - b)GCF of ax + bx is xGCF of ay + by is yx (a + b) + y(a + b) (x + y)(a + b)CHAPTER 8 MONOMIALS and POLYNOMIALS REVIEW MULTIPLYING MONOMIALS When multiplying monomials of the same base, we ADD the exponents and MULTIPLY the coefficients. See the base more than once (Left to Right) and combine the powers to make one base. Example: (4m3n4)(5m5n3) = (4*5) m3+5 n4+3 Solution: 20m8n7 DIVIDING MONOMIALS When dividing monomials of the same base, we SUBTRACT the exponents and SIMPLIFY the coefficients. Subtract the bigger MINUS smaller power and place the new base in the location of the bigger. Example:  EMBED Equation.3 ; Solution:  EMBED Equation.3   EMBED Equation.3 ; x: 5 3 = 2 in top; y: 8 2 = 6 in bottom POWERS OF MONOMIALS When we raise a monomial to an exponent, we MULTIPLY the exponents in the monomial and COEFFICIENTS gains the exponent. Example 1: (-2m2n5)3 = (-2)3 m2*3 n5*3 SOLUITION: -8 m6 n15EXAMPLE 2:  EMBED Equation.3 SPECIAL MONOMIAL CASES: A ZERO EXPONENT: CANCELS THE BASE Algebraically: If a = base, then a0 = 1 Example:  EMBED Equation.3  NEGATIVE EXPONENTS: Change the exponent to a positive and switch its location (OR bottom to top) (Top to Bottom): If a = base,  EMBED Equation.3  (Bottom to Top): If a = base, EMBED Equation.3  Examples: #1:  EMBED Equation.3  #2:  EMBED Equation.3  #3:  EMBED Equation.3 POLYNOMIAL INTRODUCTION: POLYNOMIAL: A monomial or the sum (addition or subtraction) of monomials. Degree of a monomial: the sum of the exponents of all its variables. Degree of 5mn2 is 1 + 2 = 3 Degree of a polynomial: the greatest degree of any term in the polynomial. Degree of -4x2y2 + 3x2 + 5y is 4. ADDING AND SUBTRACTING POLYNOMIALS: Combine Like Terms between polynomials. Do not change the exponents of your variable only the coefficients from your addition or subtraction. Examples: 1. (3x2 4x + 8)+(2x 7x2 5) = - 4x2 - 2x + 3 3x2 + -7x2 + -4x + 2x + 8 + -5 2. (3n2 + 13n3 + 5n)(7n + 4n3) = 9n3 + 3n2 2n 3n2 + 13n3 4n3 + 5n 7n DISTRIBUTIVE PROPERTY: Circle the ENTIRE term in front of parentheses Draw Arrows to each term in polynomial include plus or minus sign Multiply to get new terms Add all terms -3x2 (6xy2 3x + 5x2y + 2y - 8) (-3x2)(6xy2) = -18x3y2 (-3x2)( 3x) = 9x3 (-3x2)( 5x2y) = -15x4y (-3x2)( 2y) = -6x2y (-3x2)(- 8) = 24x2 Solution: -18x3y2 + 9x3 - 15x4y - 6x2y + 24x2FOIL (Arrow or Box Method) Circle the EACH term in FIRST Binomial include plus or minus sign Draw Arrows from each circled term to each term in 2nd Binomial Multiply to get 4 new terms Combine any Like Terms  (5x 3) (2x + 7) F: (5x)(2x) = 10x2 O: (5x)(7) = 35x I: (-3)(2x) = -6x L: (-3)(7) = -21 Solution: 10x2 + 29x - 21  SPECIAL PRODUCTS Special products are shortcuts for FOIL SQUARE OF A SUM: The square of a + b is the square of a plus twice the product of a and b plus the square of b.  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hV$If^V$IfG9H9I9J9M9k99999999999ʿʬҕʈʕshh\N\h4LB*CJH*aJphh4LB*CJaJphh4L5CJ$H*aJ$)jh4L5CJ$UaJ$mHnHtHuUh4L5>*CJ$aJ$h4L5CJ$aJ$jVhCJEHUaJ%ju!,O h4LCJUVaJnHtHh4L5>*CJaJh4LCJaJjh4LCJUaJj_ThCJEHUaJ%j4O h4LCJUVaJnHtHthe EACH term in FIRST polynomial include plus or minus sign Draw Arrows from each circled term to each term in polynomial Multiply to get new terms Combine Like Terms (3x 2) (6x2 + 7x - 8) (3x)(6x2) = 18x3 (-2)(6x2) = -12x2 (3x)(7x) = 21x2 (-2)(7x) = -14x (3x)(-8) = -24x (-2)(-8) = 16 Solution: 18x3+ 9x2 - 38x +16 REMINDER for BOX METHOD: If the arrows dont work for you, use the BOX METHOD to organize your multiplications. 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